If you’re preparing for a midterm, final, or qualifying-style exam in a graduate course, small misunderstandings can cascade into big errors. Online tutoring can help you identify exactly where the logic breaks and fix it quickly.
This post explains the topic as it is typically taught in graduate programs, emphasizing careful definitions, assumptions, and methodical steps. The goal is not just to state formulas, but to show how and when they apply. The discussion is designed to be accurate, self-contained, and appropriate for graduate-level work.
For guided help, see Macroeconomics Tutoring or the parent page Economics Tutoring. More worked examples are collected in the Economics Post Hub. If you’re stuck on FOCs or concavity checks, visit Troubleshooting: Theory.
Problem
Crusoe allocates labor $\ell\in(0,1)$ to production:
$$ y=f(\ell)=4\ell^{1/2}. $$
Utility is:
$$ u(c,1-\ell)=\ln c+\ln(1-\ell). $$
Since $c=y$, Crusoe solves:
$$ \max_{0<\ell<1}\; \ln(4\ell^{1/2})+\ln(1-\ell). $$
Solution
$$ U(\ell)=\ln 4+\frac{1}{2}\ln\ell+\ln(1-\ell). $$
$$ U'(\ell)=\frac{1}{2\ell}-\frac{1}{1-\ell}. $$
$$ \frac{1}{2\ell}=\frac{1}{1-\ell} \quad\Rightarrow\quad \ell^*=\frac{1}{3}. $$
$$ c^*=4\sqrt{\frac{1}{3}}=\frac{4}{\sqrt{3}}. $$
$$ U”(\ell)=-\frac{1}{2\ell^2}-\frac{1}{(1-\ell)^2}<0. $$
Leave a Reply